Capital income in a recession

Profits and income from capital are not the same thing, though they are mixed together in the national accounts. (Paul Krugman has made this point before.)

I work through some simple examples with sticky prices and/or sticky wages, where income from capital always falls in a recession caused by tight monetary policy.

[I add an incomplete postscript on how this will affect real interest rates in a recession.]

Start with a simple model.

Households own labour and capital goods. Firms pay a wage W to rent the services of labour, and pay a rental R to rent the services of capital goods, and use those services of labour and capital goods to produce consumption goods and capital goods, which they sell to households. The value of output sold, minus wages paid, minus capital rentals paid, equals firms' profits. Households own the firms, and so own those profits too.

For simplicity, assume that the supply curves of the services of the existing stocks of labour and capital goods are perfectly inelastic. Net population growth, and net investment, will make those existing stocks grow over time.

Start with full employment of both labour and capital, then assume the central bank does something stupid and there is a recession. We know that total income (equals total output) falls. But what happens to the distribution of income? That depends. On nominal rigidities.

1. Here is one extreme case. Assume output prices P are fixed, but that wages W and capital rentals R are perfectly flexible. The recession causes both W and R to fall to zero. There are unemployed workers and/or unemployed machines, both willing to work for nothing. Firms can produce output for free, but don't bother hiring the unemployed workers and machines, even at zero wages and rentals, because they can't sell the extra output.

In that extreme case, a recession causes income from wages and income from capital to fall to zero. All income is profit income. Households earn all their income from owning firms, that are able to sell some output, but not as much output as before the recession. ("But what exactly is a 'firm'?" I hear you ask. Good question.) What matters for your income in a recession is being able to sell goods, not being able to produce goods.

Now change the assumptions very slightly. Assume that firms own the capital goods they use, and pay rents to themselves. Nothing changes. We get exactly the same results as before. Except we can't tell the difference between income from profits and income from capital rentals. They both get lumped together as "profit" in the national accounts, because nobody observes the rents that firms charge themselves. But the "shadow" rental on machines -- the price that firms would be willing to pay to rent a machine -- is still zero.

2. Now let's assume that both P and W are fixed, and repeat the experiment. R is perfectly flexible. (That makes sense if firms own the capital they use, because no firm would leave its capital unemployed because it were charging itself too high a rent.)

With P and W fixed, the recession will cause unemployment of labour. It will not cause unemployment of capital, unless the technology requires labour and capital to be used in fixed proportions.

Under fixed proportions technology, capital will be unemployed, and R will drop to zero.

Under variable proportions technology, capital will be fully employed, and R will drop, but not to zero. Firms cannot sell the extra output from employing extra capital, but they can produce the same output with more capital and less labour. The cost-minimising mix of capital and labour is determined by:

MPK/MPL = R/W

(MPK and MPL are the marginal physical products of capital and labour -- the extra output per extra machine or worker employed, holding employment of the other input constant.) Since employment of labour is lower in the recession, while employment of capital in unchanged, the K/L ratio increases in a recession, so MPK/MPL falls, so R/W falls, and since W/P stays constant by assumption, R/P must fall.

3. Now let's assume that W is fixed, and that P and R are both perfectly flexible.

Under fixed proportions technology, both labour and capital will be unemployed in the recession, so R will fall to zero.

Under variable proportions technology, only labour will be unemployed, and individual firms will be able to sell as much output as they like. Profit-maximisation ensures:

MPL = W/P and MPK = R/P

Since K/L increases in a recession, MPK will fall, so R/P must fall too.

In all three cases considered above, R/P falls in a recession. Income from capital falls in a recession caused by tight monetary policy. Wage income falls in the first and second cases, and may either rise or fall in the third case. Profit income will rise in the first case, and may either rise or fall in the second and third cases (I think).

[Postscript on interest rates: Just to remind you, R is not the same as the rate of interest. It is how many dollars you pay per year to rent one machine. It is however related to the rate of interest. Let Pk be the price of capital goods, Pc be the price of consumption goods, i be the nominal rate of interest, r be the real rate of interest (deflated by Pc), and ignore depreciation, then:

i = R/Pk + Pkdot/Pk

The RHS of that equation represents the rate of return from owning a machine, which is the annual rents earned by the machine, divided by what you paid to buy the machine, plus the rate of increase of the price of machines, which will equal the nominal interest rate in equilibrium, if households are indifferent between owning machines and owning bonds. (This assumes you can actually sell your machine at the price Pk, i.e. that Pk is not sticky.)

Subtracting the inflation rate on Pc from both sides we get the real interest rate:

We only get the simple r = R/Pk = R/P = MPK in a one-good model where Pk=Pc=P, and where P is perfectly flexible.

The fall in R during a recession will tend to lower real and nominal interest rates, but the full effect on r can only be figured out if we work out what happens to Pk/Pc. In the simple one-good model, where Pk/Pc=1, we know that r will fall in a recession. In a two-good model, it will be possible for r to rise in a recession, provided Pk jumps down at the beginning of the recession, so that Pk is expected to rise faster than Pc during the recession.

Update: I think it depends on how long the recession is expected to last. If the recession is expected to last a long time, Pk/Pc will not be expected to be rising quickly, so r will fall because R falls. The IS curve slopes up. If the recession is expected to be very short, Pk/Pc will be expected to be rising quickly, so r will rise despite the fall in R. The IS curve slopes down.]

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> They both get lumped together as "profit" in the national accounts, because nobody observes the rents that firms charge themselves. But the "shadow" rental on machines -- the price that firms would be willing to pay to rent a machine -- is still zero.

We might be able to still see that in the national accounts. If a firm is unwilling to rent a machine, then it will be equally uninterested in maintaining its machinery, and investment will fall relative to the relatively constant depreciation.

The investment ratio allows us to infer that since the 2008 recession, capital just isn't worth what it used to be. The ratio remains positive if you remove change in business inventory from the gross/net investment figures, but it still approaches 0. The corporate profits:GDP ratio allows us to infer that corporate profits have in general increased (the ~10% current figure appears to be a historic high).

I don't think that's quite enough data to fully disambiguate which type of recession we've seen recently, but my hunch is that we're somewhere between 1 and 2 (prices fixed, rents fully flexible, and wages slightly flexible), with fixed and variable proportions differing by industry.

Majromax: yep. The fact that investment falls in a recession is certainly consistent with this story. It's interesting to think about investment in human capital too. The big difference between investment in human and non-human capital is that investment in human capital is mostly home production. The biggest input is your own time, and you sell the improved human machine to yourself. Only later, when the recession (you hope) is over, do you sell/rent your services to other people.

If we're going to look at investment in human capital in terms of investment, then that might also explain the perception of anemic wage growth. Home investment in human capital is hopefully repaid later in the form of higher wages.

If a firm is unwilling to rent new machinery to increase output, then with perfect information it would also be unwilling to pay higher wages to more productive (higher capital) workers to increase output, since the difference is just like renting a machine. That suggests that the marginal, aggregate return on human investment should roughly equal the marginal return on capital.

Then, a firm's desire to disinvest capital (R at or below depreciation) during a recession should also imply that a firm would seek to disinvest human capital -- hire individually less productive workers. That might(?!) go on to explain the popular perception of a "hollowing-out middle class," since we traditionally define the middle class as "professional" workers with high levels of education/training/human capital.

That could then go on to explain (good-) "jobless recoveries," since a representative firm's desire for skilled versus unskilled labour would roughly track the bold green line in that FRED chart, and the 2008-9 recession was a real killer there.

But the above is a *lot* of speculation, and real-life behaviour can't be so clean.

TMF: This is macro. Recessions are a macro topic. you can't look at one firm and ask if it's a recession.

It does not matter for this post *how* the central bank tightens monetary policy to cause a recession. All that matters is that there is a monetary policy change that causes a recession. I should definitely *not* state *how* the central bank does this. Because that would just mean that people who like to chase red herrings would chase a lot of red herrings.

Instead of assuming the CB does something stupid, why can we not think that households increase their savings? Should that not mean K rises, MPK goes down and your conclusion applies: "Income from capital falls"?

Odie: we could assume that the recession is caused by some real shock, to which the central bank responds badly. But that makes it more complicated, because we have to decompose it into two shocks: the original real shock; the central bank's bad response.

Hmm, not sure yet if my question is right or wrong. Let's say there is no CB and your assumption applies there is some real shock that makes households to save more. Would not income from capital still fall?

Looking at the onset of the most recent recession, labor share (W/O) did not rise. Thus capital share did not change. Yet, corporate profits/O were falling. So, R/O had to rise in order for the above equation to equal 1.
We then saw capital drive the recovery and corporate profits/O rebounded very quickly compared to past recessions. While labor share continued to fall.

I see your question is about distribution of income...
So is it correct? ... Income from capital "as a share of output" would not always fall during the onset of a recession?

In this case, higher real output causes higher real interest rates. Looks like an IS curve that slopes up.

But higher real interest rates don't cause higher real output.

Oh, cause? it is just equilibrium conditions right? It is the same problem as the Fisher effect discussion.

If the IS curve is considered as a demand curve, then we add up how much various firms and households will spend at different real interest rates. This is at a given level of real income.

Income and output is another thing that would influence how much they will spend too.

While real expenditure equals real output in equilibrium, I don't think it is necessary or desirable to construct the demand curves in this fashion. My spending is someone else's income, not my own income

I think the least bad way to deal with this is to shift the IS curve due to changes in income and output.

Sure, it is a bit puzzling if you focus on this current output effect on equilibrium interest rates. But there is no problem if the IS curve shifts from expected future output and income.

Your final PPS, about how a short recession leaves the IS curve negative seems related.

It is the expectation of future low output and income that reduces the demand for output at any given interest rate.

I think constructing an IS curve from a Keynesian cross is pretty pointless. And I don't use them with LM curves really. Just think of an IS curve as showing the relationship between exogenous changes in real interest rates and expenditures. And use it with a vertical potential income curve. I think illustrates a lot for an economy that is working right. Various things shift it around. If interest rates are too high or too low, there are surpluses or shortages of output.

Agreed. If you draw an upward-sloping IS curve, and a horizontal (or flatter than the IS) LM curve, you get what would reasonably be called an unstable equilibrium. But it works fine if you draw the LM curve steeper than the IS.

The underlying problem here is that the ISLM model is a one-period model. As you say yourself: "It is the expectation of future low output and income that reduces the demand for output at any given interest rate."

"If the IS curve is considered as a demand curve, then we add up how much various firms and households will spend at different real interest rates. This is at a given level of real income."

If we did draw a demand curve showing desired spending at different interest rates for a given level of real income, it would slope down. But I do not interpret the IS curve this way. I interpret it as a semi-equilibrium condition: combinations of *actual sales* (not desired sales, or supply) and interest rates such that output demanded would equal actual sales.

Here is one way to think about the IS curve: assume that people and firms have static expectations for the next n periods (they think that future income for the next n periods will be whatever it is right now), and then they expect a return to the LRAS after n periods. In other words, the recession will last for exactly n periods, with nothing changing, then disappear. Will the IS curve slope up or down? The bigger is n, the more likely it will slope up.

Alternatively, we could imagine that people expect with probability 1/n that the recession will continue just the same next period, and with probability 1-1/n will disappear forever.

> It does not matter for this post *how* the central bank tightens monetary policy to cause a recession. All that matters is that there is a monetary policy change that causes a recession.

Veering a tiny bit off-topic (my apologies), is it possible to have a monetary recession without central bank action? I'm thinking of the "financial crisis" that sparked the 2008 recession. It was seemingly caused by the sudden and unexpected downgrade of a variety of debts, which were used in a money-like fashion as slightly less-liquid mediums of exchange and as collateral for loans. If I squint hard enough, that situation seems not-entirely-distinguishable from a sudden decrease in the money supply via central bank action.

> The underlying problem here is that the ISLM model is a one-period model. As you say yourself: "It is the expectation of future low output and income that reduces the demand for output at any given interest rate."

Are there any decent micro-scale models (possibly numerical/simulation?) that end up replicating a plausible business cycle for a simplified economy? I feel (mathie intuition) that we could shed some more light on these macro questions if only we could construct a model that builds effective aggregate curves from "first principles" of micro-agents. But I'm also applying a sort of math/physics set of concepts to economics, which may (and probably would) utterly fail.

Let's see if the reasoning can go full circle by recognizing production is limited to what can be sold. R depends upon consumer income simplified to (total labor hours L * W). I assume that W determines the lion's share of consumer income. If relatively more K is employed to L, production increases faster than sales. Then Pc is suppressed. Then R is suppressed. But as technology and use of capital respond more efficiently, Pk declines raising i. As consumer income is weak in the face of normalizing production, Pcdot/Pc pulls r closer to i.
Then the question about i... is i rising or falling? Does low consumer demand depress i? Does low consumer demand depress R/Pk + Pkdot/Pk? Not necessarily. i can rise normally in spite of "excess" unemployment of labor and capital. i relates to the K in production and its price efficiency (since MPK is extra output per extra machine employed).
The utilization rates of overall capital and overall labor are independent of the efficiency of R/Pk. Workers and some capital could be marginalized in the process.
But then if a seemingly depressed economy with extra unemployed labor and capital causes one to expect a long recession to return back to full employment, then the IS curve would seem to slope up. But if those expectations don't see a demand limit upon sales and consequently production, and that some workers and capital will be left marginalized outside of the economy at "full employment", then the IS curve might reveal itself in time as sloping downward.
Hope I didn't write too much... Just some thoughts...

Majromax: "...is it possible to have a monetary recession without central bank action?"

It depends on how you define "action". What is a sin of commission, and what is a sin of omission? "Officer: it wasn't me speeding, my foot never moved on the gas pedal, it was the downhill stretch of road!"

" I feel (mathie intuition) that we could shed some more light on these macro questions if only we could construct a model that builds effective aggregate curves from "first principles" of micro-agents."

You have just described the modelling strategy of e.g. New Keynesian macro. Yes, we build models like that. You could say that what I am doing here is assuming there is some such model in the background, and asking what happens to R, and how it depends on whether P or W or both are sticky.

Edward: "Let's see if the reasoning can go full circle by recognizing production is limited to what can be sold."

That is exactly my assumption in my cases 1 and 2, where P is fixed. (And in case 3, production is limited by the amount of labour that can be sold.)

No. "Consumer" is not another word for "worker". And "consumer income" is not another word for "wage income". What you are assuming is that the marginal propensity to consume out on wage income is high (one?), and the marginal propensity to consume out of non-wage income is low (zero?). Then you need to make an assumption about the marginal propensity to invest out of non-wage income. And you need to make an assumption about what people do with their income that they do not spend on consumption or investment. Then you need to make an assumption about what the central bank does and how it affects people's consumption and investment decisions. Then you get a model of AD.

But this post is not about a model of AD. I simply assume that AD falls by some exogenous fixed amount because the central bank does something stupid. Then I ask how that fall in AD affects R, under different assumptions about the stickiness of P and W.

I was seeing a demand constraint where sales are shifted lower into the long-term. If firms cannot adjust prices as in cases 1 and 2, weak demand leads to lower sales and lower utilization of capital and labour through the long-term. LRAS shifts left. R/Pk falls. Then firms work to optimize capital within the lower sales constraint to raise R/Pk. Firms can do this, thus raising real return rates of capital in a seemingly depressed economy.
i = r + inflation = R/Pk + Pkdot/Pk.
As R/Pk rises, so does r. Yet, the twist comes as nominal interest rates are kept low because there is a concern about the new higher unemployment rate. As nominal interest rates stay low (ZLB) and real costs of financing fall below R/Pk, profits rise. Firms are content while some public needs go unmet. Firms produce more profitably for a smaller % of the population...
Yet, should the central bank never consider any tightening under these circumstances? Even if profit rates might stabilize back to traditional levels?

JKH: depreciation, in the sense of capital goods physically rusting away over time, creates a wedge between R and the rate of interest. Assuming Pk=Pc=P, for simplicity, the relation becomes: r = R/P - d , where d is the annual depreciation rate.

If capital goods wear out when being used, but not when unemployed, that will be a little different.

If capital goods become obsolete over time, that will be a little different again.

I have assumed they never rust away, never wear out, and never get obsolete. But I have allowed them to rise or fall in market value, depending on interest rates and prospective future rents.

We really ought to define capital income as R.K minus depreciation, somehow defined. It will affect my point. I don't think physical depreciation (wear or rust) or obsolescence will matter much. But market value depreciation will matter more.

Why does LRAS shift left? Lower AD causing lower utilisation of capital and/or labour (unemployed capital and/or labour) is not the same as the LRAS shifting left. It means the economy has moved to the left of the LRAS curve.

> We really ought to define capital income as R.K minus depreciation, somehow defined. It will affect my point. I don't think physical depreciation (wear or rust) or obsolescence will matter much. But market value depreciation will matter more.

I think that the two should be disentangled.

Physical depreciation puts a (negative) floor on capital income: by using but not maintaining machinery, a firm can temporarily increase its rate of production at the expense of its capital stock (in machine-dominated units). However, a firm cannot physically disassemble a machine to further increase production.

Market value depreciation matters only on the level of the individual firm. A single firm can choose to sell its machinery (or rent it out) and use the income for machine-free production. In aggregate, all firms can't do this since sellers have to be matched with buyers.

Of course, that only matters for the absolute levels of capital income. Relative measures have the market value of machinery in the denominator, which will mess up statistics.

I should have finished my comment. Assume 100 firms in the whole economy, and monetary policy remains the same. 90 of them act like Intel and lay off 1 worker. 10 of them act like Facebook and hire 1 worker. Total employment falls. Spending on goods/services can fall and/or debt defaults can happen. A recession can happen.